Well logging involves recording data related to one or more characteristics of a subterranean formation penetrated by a borehole as a function of depth. The record is called a log. Many types of logs are recorded by appropriate downhole instruments placed in a housing called a sonde. The sonde is lowered into the borehole on the end of a cable, and the parameters being logged are measured as the sonde is moved along the borehole. Data signals from the sonde are transmitted through the cable to the surface, where the log is made. FIG. 1 shows an example of a sonde 2 that measures properties of formation 4 surrounding a borehole 6 using the principles of nuclear magnetic resonance (NMR). The NMR sonde 2 includes a magnet assembly 8 and an antenna 10. The magnet assembly 8 produces a static magnetic field B0 in all regions surrounding the sonde 2, and the antenna 10 produces an oscillating magnetic field B1 that is perpendicular and superimposed on the static magnetic field B0. The NMR signal comes ad. from a small resonance volume 12 which has a radial thickness that is proportional to the magnitude of the oscillating magnetic field B1 and inversely proportional to the gradient of the static magnetic field B0. The NMR sonde 2 makes measurements by magnetically tipping the nuclear spins of protons in the formation with a pulse of the oscillating magnetic field, and then detecting the precession of the tipped particles in the resonance volume 12.
As the NMR sonde 2 traverses the borehole 6 to make measurements, it experiences random accelerations due to borehole forces acting on it. These random accelerations result in displacements of the sonde, which may adversely affect the quality of the log. To further explain this point, the resonance volume 12 generally consists of thin cylindrical shells that define a sensitive region extending along the length of the sonde 2 and having a radial thickness of about 1 millimeter. If the NMR sonde 2 moves 1 millimeter or more in the radial direction, the measurements of the T2 spin-spin relaxation times of the protons may be corrupted. Also, the time during which the nuclear spins of the protons in the formation 4 are polarized by the applied magnetic fields depend on the motion of the NMR sonde 2. If the NMR sonde 2 sticks and slips while moving along the direction of the borehole, T1 relaxation-time measurements can be compromised. In another logging mode which estimates the bound fluid volume by first saturating the nuclear spins and then letting them recover during a small time, the measurement mode overestimates the bound fluid volume if the tool moves faster than expected along the longitudinal axis of the borehole 6, or if the tool is radially displaced by more than 1 millimeter during the recovery period.
If the displacements of the sonde during the measurement interval are known, then the portions of the NMR measurements that are distorted by motions of the sonde can be identified and discarded or corrected using appropriate compensation methods. Prior art methods have used a motion detection device, such as a strain gauge, an ultrasonic range finder, an accelerometer, or a magnetometer, to detect the motions of a sonde during a logging operation. In this manner, the motion detection device is used to establish a threshold for evaluating the quality of the log. For example, U.S. Pat. No. 6,051,973 issued to Prammer discloses using accelerometers to monitor peak acceleration values of a logging tool during a measurement interval of the logging tool. The quality of the log is improved by discarding the measurements made during the period that the peak accelerations indicate that the logging tool may have been displaced by more than allowable by the extent of the sensitive region.
In one aspect, the invention is a method for determining the displacements of a logging tool during a measurement interval of the logging tool in a borehole. The method comprises obtaining a set of accelerometer signals corresponding to accelerations of the logging tool along each of three orthogonal axes of the logging tool during the measurement interval and double integrating the set of accelerometer signals to obtain corresponding displacements of the logging tool as a function of the initial velocity of the logging tool and the gravitational acceleration, wherein the initial velocity of the logging tool and the gravitational acceleration are unknown. The method further comprises assuming a set of feasible initial velocities for the logging tool. For each feasible initial velocity, the method includes estimating the gravitational acceleration, calculating the displacements of the logging tool using the feasible initial velocity and the estimated gravitational acceleration, and determining the maximum of the calculated displacements. The lower bound on the displacements of the logging tool is set to the minimum of the maximum of the calculated displacements.
In another aspect, a method for determining the displacements of a logging tool during a measurement interval of the logging tool in a borehole comprises obtaining a set of accelerometer signals corresponding to accelerations of the logging tool along each of three orthogonal axes of the logging tool during the measurement interval and calculating a tool displacement as a time-series from the accelerometer signals. The method further includes constructing a unique quadratic polynomial of time from the displacement time-series, subtracting the unique quadratic polynomial from the displacement time-series, and setting the lower bound to the maximum of the remainder of the displacement time-series.
Other aspects and advantages of the invention will be apparent from the following description and the appended claims.